Occurrences of spurious crests on the troughs of large,
relatively steep second-order Stokes waves are anomalous and not an
inherent characteristic of real waves. Here, the effects of such
occurrences on the statistics described by the standard second-order
stochastic model are examined theoretically and by way of
simulations. Theoretical results and simulations indicate that when
spurious occurrences are sufficiently large, the standard model leads
to physically unrealistic surface features and inaccuracies in the
statistics of various surface features, in particular, the troughs and
thus zero-crossing heights of large waves. Whereas inaccuracies can
be fairly noticeable for long-crested waves in both deep and
shallower depths, they tend to become relatively insignificant in
directional waves.